| Qualitative features of periodic solutions of KdV - T. Kappeler In this paper we prove new qualitative features of solutions of KdV on the circle. The first result says that the Fourier coefficients of a solution of KdV in Sobolev space $H^N,\, N\geq 0$, admit a WKB type expansion up to first order with strongly oscillating phase factors defined in terms of the KdV frequencies. The second result provides estimates for the approximation of such a solution by trigonometric polynomials of sufficiently large degree. Downloads: 3 | |

| On normalized differentials on families of curves of infinite genus - T. Kappeler We construct normalized differentials on families of curves of infinite genus. Such curves are used to investigate integrable PDE's such as the focusing Nonlinear Schr{\"o}dinger equation. Downloads: 5 | |

| Relative torsion - D. Burghelea This paper achieves, among other things, the following: 1)It frees the main result of [BFKM] from the hypothesis of determinant class and extends this result from unitary to arbitrary representations. 2)It extends (and at the same times provides a new proof of) the main result of Bismut and Zhang [BZ] from finite dimensional representations of $\Gamma$ to representations on an ${\cal A}-$Hilbert module of finite type (${\cal A}$ a finite von Neumann algebra)... Downloads: 4 | |

| Relative torsion for representations in finite type Hilbert modules - D. Burghelea For a closed manifold equipped with a Riemannian metric, a triangulation, a representation of its fundamental group on an Hilbert module of finite type (over of finite von Neumann algebra), and a Hermitian structure on the flat bundle associated to the representation, one defines a numerical invariant, the relative torsion. The relative torsion is a positive real number and unlike the analytic torsion or the Reidemeister torsion, which are defined only when the pair manifold- representation is o... Downloads: 4 | |

| Asymptotics of spectral quantities of SchrĂ¶dinger operators - T. Kappeler In this paper we provide new asymptotic estimates of the Floquet exponents of Schr\"odinger operators on the circle. By the same techniques, known asymptotic estimates of various others spectral quantities are improved. Downloads: 18 | |

| Torsions for manifolds with boundary and glueing formulas - D. Burghelea We extend the definition of analytic and Reidemeister torsion from closed compact Riemannian manifolds to compact Riemannian manifolds with boundary $(M, \partial M)$, given a flat bundle $\Cal F$ of $\Cal A$-Hilbert modules of finite type and a decomposition of the boundary $\partial M =\partial_- M \cup \partial_+ M$ into disjoint components. In particular we extend the $L-2$ analytic and Reidemeister torsions to compact manifolds with boundary... Downloads: 5 | |

| Analyticity of Riemannian exponential maps on ${\rm Diff}(\T)$ - T. Kappeler We study the exponential maps induced by Sobolev type right-invariant (weak) Riemannian metrics of order $k\ge1$ on the Lie group of smooth, orientation preserving diffeomorphisms of the circle. We prove that each of them defines an {\em analytic} Fr\'echet chart of the identity. Downloads: 4 | |

| Generic non-selfadjoint Zakharov-Shabat operators - T. Kappeler In this paper we develop tools to study families of non-selfadjoint operators $L(\varphi), \varphi \in P$, characterized by the property that the spectrum of $L(\varphi)$ is (partially) simple. As a case study we consider the Zakharov-Shabat operators $L(\varphi)$ appearing in the Lax pair of the focusing NLS on the circle. The main result says that the set of potentials $\varphi $ of Sobolev class $H^N, N \geq 0$, so that all small eigenvalues of $L(\varphi)$ are simple, is path connected and d... Downloads: 5 | |

| On the symplectic phase space of KdV - T. Kappeler We prove that the Birkhoff map $\Om$ for KdV constructed on $H^{-1}_0(\T)$ can be interpolated between $H^{-1}_0(\T)$ and $L^2_0(\T)$. In particular, the symplectic phase space $H^{1/2}_0(\T)$ can be described in terms of Birkhoff coordinates. As an application, we characterize the regularity of a potential $q\in H^{-1}(\T)$ in terms of the decay of the gap lengths of the periodic spectrum of Hill's operator on the interval $[0,2]$. Downloads: 8 | |

| Smooth Lyapunov 1-forms - M. Farber We find conditions which guarantee that a given flow on a closed smooth manifold admits a smooth Lyapunov one-form lying in a prescribed de Rham cohomology class. These conditions are formulated in terms of Schwartzman's asymptotic cycles of the flow. Downloads: 8 | |

| Isospectrality and heat content - M. van den Berg We present examples of isospectral operators that do not have the same heat content. Several of these examples are planar polygons that are isospectral for the Laplace operator with Dirichlet boundary conditions. These include examples with infinitely many components. Other planar examples have mixed Dirichlet and Neumann boundary conditions. We also consider Schr\"{o}dinger operators acting in $L^2[0,1]$ with Dirichlet boundary conditions, and show that an abundance of isospectral deformations ... Downloads: 71 | |

| Solutions of mKdV in classes of functions unbounded at infinity - T. Kappeler In 1974 P. Lax introduced an algebro-analytic mechanism similar to the Lax L-A pair. Using it we prove global existence and uniqueness for solutions of the initial value problem for mKdV in classes of smooth functions which can be unbounded at infinity, and may even include functions which tend to infinity with respect to the space variable. Moreover, we establish the invariance of the spectrum and the unitary type of the Schr{\"o}dinger operator under the KdV flow and the invariance of the spec... Downloads: 3 | |

| Lyapunov 1-forms for flows - M. Farber In this paper we find conditions which guarantee that a given flow $\Phi$ on a compact metric space $X$ admits a Lyapunov one-form $\omega$ lying in a prescribed \v{C}ech cohomology class $\xi\in \check H^1(X;\R)$. These conditions are formulated in terms of the restriction of $\xi$ to the chain recurrent set of $\Phi$. The result of the paper may be viewed as a generalization of a well-known theorem of C... Downloads: 4 | |